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        统计学习 | 最小二乘支持向量回归与核岭回归
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    <p>最小二乘支持向量回归（Least Square Support Vector Regression, LSSVR）可以认为是支持向量回归（Support Vector Regression, SVR）的一种变体，LSSVR也等价于核岭回归（Kernel Ridge Regression，KRR）。下面我们对LSSVR与KRR进行介绍。</p>
<h2 id="1-最小二乘支持向量回归">1. 最小二乘支持向量回归</h2>
<p>LSSVM的原始问题如下：</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi>min</mi><mo>⁡</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mspace width="1em"/><mfrac><mn>1</mn><mn>2</mn></mfrac><mi mathvariant="normal">∥</mi><mi>w</mi><msubsup><mi mathvariant="normal">∥</mi><mn>2</mn><mn>2</mn></msubsup><mo>+</mo><mfrac><mi>C</mi><mn>2</mn></mfrac><msup><mi>ξ</mi><mi>T</mi></msup><mi>ξ</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">t</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mspace width="1em"/><mi>y</mi><mo>−</mo><mi>X</mi><mi>w</mi><mo>−</mo><mi>b</mi><mo>−</mo><mi>ξ</mi><mo>=</mo><mn>0</mn></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
\min &amp;\quad \frac{1}{2}\Vert w \Vert^{2}_{2} + \frac{C}{2}\xi^{T}\xi \\
\mathrm{s.t}&amp;\quad y - Xw - b - \xi = 0
\end{aligned}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.84633em;vertical-align:-1.673165em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.173165em;"><span style="top:-4.173165em;"><span class="pstrut" style="height:3.3603300000000003em;"></span><span class="mord"><span class="mop">min</span></span></span><span style="top:-2.3471650000000004em;"><span class="pstrut" style="height:3.3603300000000003em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">s</span><span class="mord mathrm">.</span><span class="mord mathrm">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.673165em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.173165em;"><span style="top:-4.173165em;"><span class="pstrut" style="height:3.3603300000000003em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:1em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord">∥</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mord"><span class="mord">∥</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em;">C</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04601em;">ξ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span></span><span class="mord mathdefault" style="margin-right:0.04601em;">ξ</span></span></span><span style="top:-2.3471650000000004em;"><span class="pstrut" style="height:3.3603300000000003em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:1em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault">b</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault" style="margin-right:0.04601em;">ξ</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.673165em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<p>相比SVR，LSSVR将松弛项变为了平方的形式，并对原来的约束进行了修改，从SVR的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">ϵ</span></span></span></span>-insensitive 损失变成了最小二乘损失，失去了原有的稀疏性，但是在求解过程上有所简化，可以变为求解线性方程组的形式。下面我们引入拉格朗日函数（Lagrange function）将约束优化问题改写为如下无约束优化问题，我们得到：</p>

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